Planetary science is certainly one of the most interesting fields of astronomy, sure it can have a bit too much in the way of exogeology which is clearly squishy, but it's exciting as hell when it comes to the information we can glean about extra-solar planets from observational data.
Now the ways we obtain data about potential systems other than our own all require some pretty damn fine measurements, listed below are some of the methods currently in use:
The Transit Method - Based around observing changes in a stars visual brightness as a planet transits the face of the stars disk. The problem with this method is primarily that it only allows us to find planets with orbits edge on to us, which accounts for only a mere 10% or so of possibilities. Additionally it's fairly easy to get a false positive from this method, and so confirmation from other methods must subsequently be gained.
Doppler Spectroscopy - This method involves the measurement of the stars radial velocity through Doppler shift due the star 'wobbling' because of the gravitational influence of a nearby planet or planets. Thanks to recent advances in modern spectrometers this is by far the most productive method currently in use, and has allowed us to catalog thousands of extra-solar planets. A drawback for the moment is that we have so far only been able to detect massive planets in close orbits, this is because measurements must be taken over a period of time directly proportional to the orbital period of the planet in question. Basically we've just not been observing long enough yet!
Infra-red imaging of Circumstellar disks - Protoplanetary disks around a star will tend to absorb ordinary starlight and emit it back out into space as infrared radiation which can usually be detected, even in relatively small quantities. The presence of a formed planet can sometimes be inferred from this due to observed gaps in the disk itself that may be caused by the gravitational influence of a planet clearing the material in its orbital path.
Gravitational microlensing - By far the coolest method listed here. Entirely dependant on having a background star almost exactly inline with the system you wish to observe. It works by detecting fluctuations in the lensing effect of the parent star when a planet is positioned in such a way as to cause measurable changes. Drawbacks? Primarily that this method requires highly improbable alignment and thus a large number of sample subjects need to be monitored at the same time.
Or, just directly observe em forming - The pretty way, just check out those Proplyds!
However you look at it, some pretty ingenious methods have been devised in our quest for other planets, certainly not all of which have been listed here. Driven by the burning question as we are - are we alone? - many amazing discoveries in this field still await us.